Cooperative coloring of some graph families

TL;DR

This paper investigates the minimum number of graphs needed to guarantee a cooperative coloring in various graph classes with bounded degree, providing exact values for trees and wheels and asymptotic bounds for more complex classes.

Contribution

It introduces the concept of cooperative coloring for graph families and determines exact and asymptotic bounds for specific graph classes with degree constraints.

Findings

Exact value m_T(3)=4 for trees.

Exact value m_W(4)=5 for wheels.

Asymptotic bounds for bipartite and theta graphs.

Abstract

In a family ${G_1, G_2, \ldots, G_m}$ of graphs sharing the same vertex set $V$, a cooperative coloring involves selecting one independent set $I_i$ from $G_i$ for each $i\in \{1,2,\ldots,m\}$ such that $\bigcup_{i=1}^m I_i = V$. For a graph class $\mathcal{G}$, let $m_{\mathcal{G}}(d)$ denote the minimum $m$ required to ensure that any graph family ${G_1, G_2, \ldots, G_m}$ on the same vertex set, where $G_i\in\mathcal{G}$ and $\Delta(G_i)\leq d$ for each $i\in \{1,2,\ldots,m\}$, admits a cooperative coloring. For the graph classes $\mathcal{T}$ (trees) and $\mathcal{W}$ (wheels), we find that $m_\mathcal{T}(3)=4$ and $m_\mathcal{W}(4)=5$. Also, we prove that $m_{\mathcal{B}^*}(d)=O(\log_2 d)$ and $m_{\mathcal{L}}(d)=O\left(\frac{\log d}{\log\log d}\right)$, where $\mathcal{B}^*$ represents the class of graphs whose components are balanced complete bipartite graphs, and $\mathcal{L}$ represents the class of graphs whose components are generalized theta graphs.